Let $X$ be a CW complex. If $E$ is a vector bundle over $X$, then it's well-known that the Stiefel-Whitney classes $w_j(E) \in H^j(X,\mathbb F_2)$ of $E$ are determined from the classes $w_{2^k}(E)$ (for $2^k \leq j$) via the Wu formula, using the cup product and the action of the Steenrod algebra.

Question 1: Does the Wu formula imply any further relations? This is a purely algebraic question which I make more precise in (a) and (b) below.

That is, let $H$ be a nonnegatively-graded $\mathbb F_2$-algebra with an unstable action of the Steenrod algebra satisfying the Cartan formula and $Sq^{|x|}(x) = x^2$ for all homogenenous $x \in H$. Let $W$ be the set of sequences $(w_j \in H^j)_{j \in \mathbb N}$ with $w_0 = 1$ and satisfying the Wu formla.

(a) For any sequence $(v_{2^k} \in H^{2^k})_{k \in \mathbb N}$, does there exist $w \in W$ (necessarily unique) with $w_{2^k} = v_{2^k}$ for all $k \in \mathbb N$?

Presumably (i) the Whitney sum formula and (ii) the universal formula for the Stiefel-Whitney classes of a tensor product of vector bundles are compatible with the Wu formula, so that $W$ is a commutative ring using (i) for addition and (ii) for multiplication.

(b) Is $W$ a polynomial algebra on whichever generators from (a) do exist?

Question 2: What restrictions beyond the Wu formula are there restricting the Stiefel-Whitney classes of a vector bundle $E$ on a CW complex $X$? This is a genuinely topological question.

Over here Mark Grant describes one such restriction, but ideally I'd like a more systematic discussion.

If it simplifies matters to assume that $X$ is finite, or even a compact manifold, then that's fine.

  • $\begingroup$ I once asked a related question: mathoverflow.net/questions/239482/… $\endgroup$ – Jens Reinhold Jul 3 '20 at 6:53
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    $\begingroup$ Isn't Question 2 answered by the integral cohomology of $BO(n)$? $\endgroup$ – John Greenwood Jul 3 '20 at 15:55
  • $\begingroup$ @JohnGreenwood I'm not sure what you mean? I agree that the integral cohomology of $BO(n)$ will pull back to "integral characteristic classes" which probably contain some information that the Stiefel-Whitney classes don't. Do you have something more specific in in mind? $\endgroup$ – Tim Campion Jul 3 '20 at 18:34

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